Recent experiments have revealed that a variety of associative polymers with different architecture (linear and branched) and different nature of the associating interaction (associative protein domains and metal-ligand bonds) exhibit unexplained superdiffusive behavior. Here, Brownian dynamics simulations of unentangled coarse-grained associating star shaped polymers are used to establish a molecular picture of chain dynamics that explains this behavior. Polymers are conceptualized as particles with effective Rouse diffusivities that interact with a mean field background through attachments by stickers at the end of massless springs that represent the arms of the polymer. The simulations reveal three mechanisms of molecular diffusion at length scales much larger than the radius of gyration: hindered diffusion, walking diffusion, and molecular hopping, all of which depend strongly on polymer concentration, arm length, and the association/dissociation rate constants. The molecular model establishes that superdiffusive scaling results primarily from molecular hopping, which only occurs when the kinetics of attachment are slower than the relaxation time of dangling strands. Scaling relationships can be used to identify the range of rate constants over which this behavior is expected. The formation of loops in the networks promotes this superdiffusive scaling by reducing the total number of arms that must detach in order for a hopping step to occur.